(随机漫步模型--统计学)RandomWalkModels.ppt

1、Statistics and Data Analysis,Professor William Greene Stern School of Business Department of IOMS Department of Economics,Statistics and Data Analysis,Random Walk Modelsfor Stock Prices,A Model for Stock Prices,Preliminary: Consider a sequence of T random outcomes, independent from one to the next,

2、1, 2, T. ( is a standard symbol for “change” which will be appropriate for what we are doing here. And, well use “t” instead of “i” to signify something to do with “time.”) t comes from a normal distribution with mean and standard deviation .,1/30,Application,Suppose P is sales of a store. The accou

3、nting period starts with total sales = 0 On any given day, sales are random, normally distributed with mean and standard deviation . For example, mean $100,000 with standard deviation $10,000 Sales on any given day, day t, are denoted t 1 = sales on day 1, 2 = sales on day 2, Total sales after T day

4、s will be 1+ 2+ T Therefore, each t is the change in the total that occurs on day t.,2/30,Using the Central Limit Theorem to Describe the Total,Let PT = 1+ 2+ T be the total of the changes (variables) from times (observations) 1 to T. The sequence is P1 = 1 P2 = 1 + 2 P3 = 1 + 2 + 3 And so on PT = 1

5、 + 2 + 3 + + T,3/30,Summing,If the individual s are each normally distributed with mean and standard deviation , then P1 = 1 = Normal , P2 = 1 + 2 = Normal 2, 2 P3 = 1 + 2 + 3= Normal 3, 3 And so on so that PT = NT, T,4/30,Application,Suppose P is accumulated sales of a store. The accounting period

6、starts with total sales = 0 1 = sales on day 1, 2 = sales on day 2 Accumulated sales after day 2 = 1+ 2 And so on,5/30,This defines a Random Walk,The sequence is P1 = 1 P2 = 1 + 2 P3 = 1 + 2 + 3 And so on PT = 1 + 2 + 3 + + T,It follows that P1 = 1 P2 = P1 + 2 P3 = P2 + 3 And so on PT = PT-1 + T,6/3

7、0,A Model for Stock Prices,Random Walk Model: Todays price = yesterdays price + a change that is independent of all previous information. (Its a model, and a very controversial one at that.) Start at some known P0 so P1 = P0 + 1 and so on. Assume = 0 (no systematic drift in the stock price).,7/30,Ra

8、ndom Walk Simulations,Pt = Pt-1 + t Example: P0= 10, t Normal with =0, =0.02,8/30,Uncertainty,Expected Price = EPt = P0+TWe have used = 0 (no systematic upward or downward drift). Standard deviation = T reflects uncertainty. Looking forward from “now” = time t=0, the uncertainty increases the farthe

9、r out we look to the future.,9/30,Using the Empirical Rule to Formulate an Expected Range,10/30,Application,Using the random walk model, with P0 = $40, say =$0.01, =$0.28, what is the probability that the stock will exceed $41 after 25 days? EP25 = 40 + 25($.01) = $40.25. The standard deviation will

10、 be $0.2825=$1.40.,11/30,Prediction Interval,From the normal distribution,Pt - 1.96t X t + 1.96t = 95% This range can provide a “prediction interval, where t = P0 + t and t = t.,12/30,Random Walk Model,Controversial many assumptions Normality is inessential we are summing, so after 25 periods or so,

11、 we can invoke the CLT. The assumption of period to period independence is at least debatable. The assumption of unchanging mean and variance is certainly debatable. The additive model allows negative prices. (Ouch!) The model when applied is usually based on logs and the lognormal model. To be cont

12、inued ,13/30,Lognormal Random Walk,The lognormal model remedies some of the shortcomings of the linear (normal) model. Somewhat more realistic. Equally controversial. Description follows for those interested.,14/30,Lognormal Variable,If the log of a variable has a normal distribution, then the varia

13、ble has a lognormal distribution. Mean =Exp+2/2 Median = Exp,15/30,Lognormality Country Per Capita Gross Domestic Product Data,16/30,Lognormality Earnings in a Large Cross Section,17/30,Lognormal Variable Exhibits Skewness,The mean is to the right of the median.,18/30,Lognormal Distribution for Pric

14、e Changes,Math preliminaries: (Growth) If price is P0 at time 0 and the price grows by 100% from period 0 to period 1, then the price at period 1 is P0(1 + ). For example, P0=40; = 0.04 (4% per period); P1 = P0(1 + 0.04). (Price ratio) If P1 = P0(1 + 0.04) then P1/P0 = (1 + 0.04). (Math fact) For sm

15、allish , log(1 + ) Example, if = 0.04, log(1 + 0.04) = 0.39221.,19/30,Collecting Math Facts,20/30,Building a Model,21/30,A Second Period,22/30,What Does It Imply?,23/30,Random Walk in Logs,24/30,Lognormal Model for Prices,25/30,Lognormal Random Walk,26/30,Application,Suppose P0 = 40, =0 and =0.02. W

16、hat is the probabiity that P25, the price of the stock after 25 days, will exceed 45? logP25 has mean log40 + 25 =log40 =3.6889 and standard deviation 25 = 5(.02)=.1. It will be at least approximately normally distributed. PP25 45 = PlogP25 log45 = PlogP25 3.8066 PlogP25 3.8066 = P(logP25-3.6889)/0.

17、1 (3.8066-3.6889)/0.1)=PZ 1.177 = PZ -1.177 = 0.119598,27/30,Prediction Interval,We are 95% certain that logP25 is in the intervallogP0 + 25 - 1.9625 to logP0 + 25 + 1.9625. Continue to assume =0 so 25 = 25(0)=0 and =0.02 so 25 = 0.02(25)=0.1 Then, the interval is 3.6889 -1.96(0.1) to 3.6889 + 1.96(

18、0.1)or 3.4929 to 3.8849.This means that we are 95% confident that P0 is in the rangee3.4929 = 32.88 and e3.8849 = 48.66,28/30,Observations - 1,The lognormal model (lognormal random walk) predicts that the price will always take the form PT = P0et This will always be positive, so this overcomes the problem of the firs